What is the integral of (x^3-3x^2+2)/(x-2) ?

The integral of (x^3-3x^2+2)/(x-2) is 3x^2-12x+1/3 (x-2)^3-2ln|x-2|-3/2 (x-2)^2+12+C

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integral of (x^3-3x^2+2)/(x-2)

\int \frac{x ^{3} - 3 x ^{2} + 2}{x - 2} d x

3 x^{2} - 12 x + \frac{1}{3} (x - 2)^{3} - 2 ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 12 + C

Solution steps

\int \frac{x ^{3} - 3 x ^{2} + 2}{x - 2} d x

Expand

\frac{x ^{3} - 3 x ^{2} + 2}{x - 2} : \frac{x ^{3}}{x - 2} - \frac{3 x ^{2}}{x - 2} + \frac{2}{x - 2}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x - 2} d x - \int \frac{3 x ^{2}}{x - 2} d x + \int \frac{2}{x - 2} d x

\int \frac{x ^{3}}{x - 2} d x = 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12

\int \frac{3 x ^{2}}{x - 2} d x = 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

= 3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣

Simplify

3 x^{2} + 8 ln ∣ x - 2 ∣ + \frac{1}{3} (x - 2)^{3} - 12 - 3 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) + 2 ln ∣ x - 2 ∣ : 3 x^{2} - 12 x + \frac{1}{3} (x - 2)^{3} - 2 ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 12

= 3 x^{2} - 12 x + \frac{1}{3} (x - 2)^{3} - 2 ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 12

Add a constant to the solution

= 3 x^{2} - 12 x + \frac{1}{3} (x - 2)^{3} - 2 ln ∣ x - 2 ∣ - \frac{3}{2} (x - 2)^{2} + 12 + C

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\int \frac{\partial}{\partial x} (x^{2}) d x

What is the integral of (x^3-3x^2+2)/(x-2) ?
The integral of (x^3-3x^2+2)/(x-2) is 3x^2-12x+1/3 (x-2)^3-2ln|x-2|-3/2 (x-2)^2+12+C